
Counting triples relative to a fixed line.
Let \(l_0\in L\) . We claim that
\(\begin{split}\#& \lbrace (p,l,q)\in I(l_0) \times L \times P : (p,l),(q,l) \in I; \operatorname{dist}(p, q)\gtrapprox 1; \angle (l_0,l)\ge \delta ^\varepsilon ;\\& \textrm {there are}\ \gtrapprox \delta ^{-\alpha }V\ \textrm {points incident to}\ l\ \textrm {between}\ p\ \textrm {and}\ q\rbrace  \\&\gtrapprox \Big (\delta ^{-\alpha }V\Big )\Big (\delta ^{-\alpha }(\#L)(\#P)^{-1}V\Big )\Big (\delta ^{-\alpha }V\Big )\\&= \delta ^{-3\alpha }(\#L)(\#P)^{-1}V^2.\end{split}\) 
The set in (REF ) is constructed as follows: first we select a point \(p\in I(l_0)\) ; by (REF ) there are \(\gtrapprox \delta ^{-\alpha }V\)  choices for \(p\) . Then we select a line \(l\in I(p)\)  with \(\angle (l_0,l)\ge \delta ^\varepsilon \) ; by (REF ) and (REF ), there are \(\gtrapprox \frac{(\#I)}{(\#P)}\approx \delta ^{-\alpha }(\#L)(\#P)^{-1}V\)  choices for \(l\) . Finally we select a point \(q\in I(l)\)  with \(\operatorname{dist}(p, q)\gtrapprox 1\)  and for which there are \(\gtrapprox \delta ^{-\alpha }V\ \textrm {points incident to}\ l\ \textrm {between}\ p\ \textrm {and}\ q\) . By (REF ) and (REF ), there are \(\gtrapprox \delta ^{-\alpha }V\)  choices for \(q\) .
For each \(q\in P\) , let \(L(q)\subset L\)  be the set of lines \(l\)  so that there exists \(p\in P\)  such that \((p,l,q)\in (\ref {triplesplq})\) . For each \(l\in L(q)\) , there are \(\lessapprox 1\)  points \(p\in P\)  so that \((p,l,q)\in (\ref {triplesplq})\)  (indeed, such a point \(p\)  must be contained in \(N_{\delta }(l)\cap N_{\delta }(l_0)\) , and these two lines intersect at angle \(\approx 1\) ).

Applying a two-ends reduction.
Next we will apply a two-ends reduction; see [1]} for an introduction to this topic. Fix \(q\in P\) , and for each \(r\in [\delta ,\pi ]\)  and \(v\in S^1\) , define
\(f(r,v)=r^{-\varepsilon }\#\lbrace l\in L(q)\colon \angle (l, v) \le r\rbrace .\) 
This function is upper semi-continuous (indeed, for every \((r,v)\in [\delta ,\pi ]\times S^1\) , there exists \(\tau >0\)  so that \(f(r^{\prime },v^{\prime })\le f(r,v)\)  for all \(|r-r^{\prime }|<\tau ,\ |v-v^{\prime }|<\tau \) ), and thus achieves its maximum on the compact set \([\delta ,\pi ]\times S^1\) . Let \((r_q,v_q)\)  be a pair that achieves this maximum.
Define
\(L^{\prime }(q) = \lbrace l\in L(q)\colon \angle (l_i,v_q)<r\rbrace .\) 
Since \(f(\pi , v)=\pi ^{-\varepsilon }\#L(q)\)  for any \(v\in S^1\) , we have \(f(r_q,v_q)\ge f(\pi , v_q) \ge \pi ^{-\varepsilon }\#L(q)\) , and hence \(\#L^{\prime }(q)\ge (r_q/\pi )^{\varepsilon }\#L(q)\ge (\delta /\pi )^{\varepsilon }\#L(q)\) .
We claim that for each \(\delta \le r\le r_q\) , there at most \((r/r_q)^\varepsilon (\# L^{\prime }(q))^4\)  quadruples \((l_1,\ldots ,l_4) \in (L^{\prime }(q))^4\)  so that \(\angle (l_3, l_4)<r\) . Indeed; once \(l_1,l_2,l_3\)  have been selected, \(l_4\)  must satisfy \(\angle (l_3,l_4)\le r\) , and hence the number of such \(l_4\)  is \(r^\varepsilon f(r,\operatorname{dir}(l_3))\) . Since \(f(r,\operatorname{dir}(l_3))\le f(r_q,v_q)\) , we have \(r^\varepsilon f(r,\operatorname{dir}(l_3))\le r^{\varepsilon }f(r_q,v_q)= (r/r_q)^\varepsilon \#L^{\prime }(q)\) .
An identical argument for each pair of distinct indices \(i\ne j\)  shows there there are at most \(\binom{4}{2} (r/r_q)^\varepsilon (\# L^{\prime }(q))^4\)  quadruples \((l_1,\ldots ,l_4) \in (L^{\prime }(q))^4\)  so that \(\angle (l_i,l_j)<r\)  for some \(i\ne j\) . Selecting \(r\sim r_q\)  appropriately, we have
\(\#\lbrace (l_1,\ldots ,l_4)\in (L^{\prime }(q))^4\colon \angle (l_i,l_j) \gtrsim r_q\ \textrm {for each}\ i\ne j\rbrace \gtrsim (\# L^{\prime }(q))^4 \approx (\#L(q))^4.\) 
By dyadic pigeonholing, we can refine the set on the LHS of (REF ) by a factor of \(1/|\log \delta |\) , and find a number \(d\)  such that \(r_q\sim d\)  for every point \(q\in P\)  that contributes to at least one triple from (REF ). Note that Inequality (REF ) remains true for this (slightly) smaller set of triples.
Finally, recall that at the beginning of this section we fixed a choice of \(l_0\in L\) . We write \(d=d_{l_0}\)  to emphasize that the value of \(d\)  determined above depends on \(l_0\) .

Applying Hölder's inequality.
Define
\(\begin{split}\mathcal {N}_{l_0} = &\lbrace (p_1,\ldots ,p_4, l_1,\ldots ,l_4, q)\in I(l_0)^4\times L^4\times P :\\& (p_i,l_i),(q,l_i)\in I;\ \operatorname{dist}(p_i,q) \approx 1;\ \angle (l_0,l_i)\ge \delta ^{\varepsilon },\ i=1,\ldots ,4;\\& \qquad \operatorname{dist}(p_1,p_2) \sim \operatorname{dist}(p_2,p_3) \sim \operatorname{dist}(p_3,p_4) \sim d_{l_0} \rbrace ,\end{split}\) 
where in the above set we chose the labeling so that \(p_1,\ldots ,p_4\)  appear in that order along \(l_0\) . Applying Hölder's inequality to the refinement of (REF ) described above, we have
\(\mathcal {N}_{l_0} \gtrapprox \delta ^{-12\alpha }(\#L)^4(\#P)^{-7}V^8.\) 
Note that if \((p_1,\ldots ,p_4, l_1,\ldots ,l_4, q)\in \mathcal {N}_{l_0}\) , then \(\operatorname{dist}(l_0,q)\approx 1\) . By (REF ), we also have that there are \(\gtrapprox \delta ^{-\alpha }V\)  points \(p_5\in P\)  with \((p_5,l_3)\in I\)  so that \(p_5\)  is between \(p_3\)  and \(q\)  on the line \(l_3\) . Thus by (REF ) and the fact that \(\angle (l_0,l_3)\approx 1\) , we have that there are \(\gtrapprox \delta ^{-\alpha } V\)  points \(p_5\in P\)  so that (i) \((p_5,l_3)\in I\) , (ii) \(\operatorname{dist}(l_0,p_5)\approx 1\) , (iii) \(\operatorname{dist}(p_5, q)\approx 1\) , and (iv) \(p_5\)  is contained in the line segment between \(p_3\)  and \(q\)  (i.e. \(p_5\)  is contained inside the triangle \(\Delta _{p_1,p_4,q}\)  spanned by \(p_1,p_4,q\) ). Items (i)–(iv) also imply that
\(\begin{split}\operatorname{dist}(q, \overline{p_1p_5})\gtrapprox d_{l_0},\quad \textrm {and}\quad \operatorname{dist}(q, \overline{p_4p_5})\gtrapprox d_{l_0},\end{split}\) 
where \(\overline{p_1p_5}\)  denotes the line spanned by \(p_1\)  and \(p_5\) , and similarly for \(\overline{p_4p_5}\) . Thus,
\(\begin{split}&\#\lbrace (p_1,\ldots ,p_4, l_1,\ldots ,l_4, q,p_5)\in \mathcal {N}_{l_0}\times P : (p_5,l_3)\in I;\ \operatorname{dist}(l_0,p_5)\approx 1;\\&\qquad \operatorname{dist}(p_5,q)\approx 1;\ p_5\in \Delta _{p_1,p_4,q};\ \operatorname{dist}(p, \overline{p_1p_5})\gtrapprox d_{l_0};\ \operatorname{dist}(p, \overline{p_4p_5})\gtrapprox d_{l_0}\rbrace \\& \gtrapprox \delta ^{-13\alpha }(\#L)^4(\#P)^{-7}V^9.\end{split}\) 

Summing over the lines.
Recall that the set described in (REF ) depends on the choice of line \(l_0\) . Let \(\mathcal {D}\)  be the disjoint union of such sets over all lines \(l_0\in L\) , i.e.
\(\begin{split}\mathcal {D} = \lbrace &(l_0,p_1,\ldots ,p_4, l_1,\ldots ,l_4, q,p_5)\colon (p_1,\ldots ,p_4, l_1,\ldots ,l_4, q,p_5)\in \mathcal {N}_{l_0}\times P;\ (p_5,l_3)\in I;\\&\operatorname{dist}(l_0,p_5)\approx 1;\ \operatorname{dist}(p_5,q)\approx 1;\ p_5\in \Delta _{p_1,p_4,q};\ \operatorname{dist}(p, \overline{p_1p_5})\gtrapprox d_{l_0};\ \operatorname{dist}(p, \overline{p_4p_5})\gtrapprox d_{l_0}\rbrace .\end{split}\) 
By (REF ), we have
\(\#\mathcal {D}\gtrapprox \delta ^{-13\alpha }(\#L)^5(\#P)^{-7}V^9.\) 
After dyadic pigeonholing, we can suppose there is a number \(d\)  so that so that \(d_{l_0}\sim d\)  for a \(|\log \delta |^{-1}\)  fraction of the tuples in \(\mathcal {D}\) . Abusing notation, we will continue to refer to this refined set as \(\mathcal {D}\) .

Fixing four special points.
Next, we will use pigeonholing to select a choice of \((l_0,p_1,p_2,p_4,p_5)\)  that occur in many tuples from \(\mathcal {D}\) . There are \(\#L\)  choices for \(l_0\) . Once \(l_0\)  has been specified, there are at most \(\delta ^{-\alpha -\varepsilon }V\)  choices for \(p_1\) . Next, there are \(\lessapprox d^\alpha \delta ^{-\alpha -\varepsilon }V\)  choices for each of \(p_2\)  and \(p_4\) . Finally, there are \(\le \#P\)  choices for \(p_5\) . Thus there is a choice of \((l_0,p_1,p_2,p_4,p_5)\)  that appears in
\(\gtrapprox \frac{ \delta ^{-13\alpha }(\#L)^5(\#P)^{-7}V^9 }{(\#L)(\delta ^{-\alpha }V)(d^\alpha \delta ^{-\alpha }V)^2(\#P) }= d^{-2\alpha }\delta ^{-10\alpha }(\#L)^4(\#P)^{-8}V^6\) 
tuples from \(\#\mathcal {D}\) . Fix this choice of \(l_0,p_1,p_2,p_4,p_5\) . Note that the point \(p_3\)  and line \(l_3\)  are determined up to multiplicity \(\approx 1\) .
Write \(p\sim q\)  if there is a line \(l\in L\)  with \((p,l),(q,l)\in I\) . Since \(\operatorname{dist}(p_i,q)\approx 1,\ i=1,2,4,5\) , if \(p_i\sim q\)  then there are \(\approx 1\)  lines \(l_i\in L\)  incident to both \(p_i\)  and \(q\) . Thus if we define
\(\begin{split}Q = \lbrace  q \in P \colon p_1\sim q;\ p_2\sim & q;\ p_4\sim q;\ p_5\sim q;\ \operatorname{dist}(l_0, q)\approx 1;\ \operatorname{dist}(p_5,q)\approx 1;\\& p_5\in \Delta _{p_1,p_4,q};\ \operatorname{dist}(q, \overline{p_1p_5})\gtrapprox d;\ \operatorname{dist}(q, \overline{p_4p_5})\gtrapprox d \rbrace ,\end{split}\) 
then
\(\#Q \gtrapprox d^{-2\alpha }\delta ^{-10\alpha }(\#L)^4(\#P)^{-8}V^6.\) 

Rescaling.
If \(R\)  is a rectangle, we define \(\operatorname{dir}(R)\in S^1\)  to be the direction of the major axis of \(R\) . In practice, we will always consider rectangles with eccentricity \(>1\) , so this is well defined. The quantity \(\operatorname{dir}(R)\)  is meaningful up to uncertainty \(\pm e^{-1}\) , where \(e\)  is the eccentricity of \(R\) .
Let \(\lbrace R\rbrace \)  be a maximal set of rectangles of dimensions \(4\times 100d\) , which are centered at the midpoint of the line segment \(p_1\) —\(p_4\) , and point in \(d\) -separated directions. Note that the union of these rectangles covers \([0,1]^2\)  (and thus covers \(P\) ), and these rectangles are boundedly overlapping in the following sense: if \(x\in [0,1]^2\)  has distance \(\approx 1\)  from the midpoint of \(p_1\) —\(p_4\) , then \(\lessapprox 1\)  rectangles from this collection contain \(x\) . In particular, each point \(q\in Q\)  is contained in \(\approx 1\)  of these rectangles.
Next, observe that if \(q\in Q\)  is contained in a rectangle \(R\) , and if \(l\in L\)  with \((p_1,l),(q,l)\in I\) , then since \(\operatorname{dist}(p_1,q)\approx 1\)  and since \(p_1\)  and \(q\)  are both contained in \(R\) , there is a constant \(C_0=C_0(\alpha )\)  (which depends on the implicit constants in the above quasi-inequalities) so that if \(\delta >0\)  is selected sufficiently small, then
\(\angle (l,\operatorname{dir}(R))\le \delta ^{-C_0\varepsilon }d.\) 
The same holds true with \(p_2\)  and \(p_4\)  in place of \(p_1\) . Crucially, the same also holds true with \(p_5\)  in place of \(p_1\) —since \(p_5\in \Delta _{p_1,p_4,q}\) , we must have \(p_5\in R\) .
By dyadic pigeonholing, we can select a set \(\mathcal {R}\)  of rectangles from the collection described above, and an integer \(N\) , so that the following holds:

\(\#\Big (Q\cap \bigcup _{R\in \mathcal {R}}R\Big )\gtrapprox \# Q\)

\(N\le \#(R\cap Q)\le 2N\)  for each \(R\in \mathcal {R}\) .

The quantity \(\# \lbrace l\in I(p_1)\colon \angle (l,\operatorname{dir}(R)) \le \delta ^{-C_0\varepsilon }d\rbrace \)  is approximately the same for each \(R\in \mathcal {R}\) .

The above item holds for each of \(p_2,p_4\)  and \(p_5\)  in place of \(p_1\) .

Define \(M = \#\mathcal {R}\) . By (REF ), for each \(R\in \mathcal {R}\)  we have
\(\#(Q\cap R)\gtrapprox d^{-2\alpha }\delta ^{-10\alpha }(\#L)^4(\#P)^{-8}V^6M^{-1},\) 
while by the third item above and (REF ), (REF ), for each \(R\in \mathcal {R}\)  we have
\(\# \lbrace l\in I(p_1)\colon \angle (l,\operatorname{dir}(R))\le \delta ^{-C_0\varepsilon }d\rbrace \lessapprox \#I(p_1)M^{-1} \lessapprox \delta ^{-\alpha } (\#L)(\#P)^{-1} V M^{-1},\) 
and similarly for each of \(p_2,p_4\)  and \(p_5\)  in place of \(p_1\) .
For the remainder of the argument, we will fix a rectangle \(R\in \mathcal {R}\) . Define \(Q_1=Q\cap R\) . Define
\(L(p_1)=\lbrace l\in I(p_1)\colon \angle (l,\operatorname{dir}(R))\le \delta ^{-C_0\varepsilon }d\rbrace .\) 
Recall that \(l\in I(p_1)\)  implies that \(p_1\in N_{\delta }(l)\) . Abusing notation (but hopefully not the sensibilities of our readers), we shall slightly perturb each line \(l\in L(p_1)\)  so that \(p_1\in l\) . If \(q\in Q\)  was a point with \(q\in N_\delta (l)\)  prior to this perturbation, then \(q\in N_{2\delta }(l)\)  after the perturbation; this is harmless for our arguments. Define \(L(p_2),L(p_4)\) , and \(L(p_5)\)  similarly. The cardinality of these sets is controlled by (REF ).
Observe that for each \(q\in Q_1\) , there are \(\approx 1\)  lines \(l_1\in L(p_1)\)  (resp. \(l_2\in L(p_2)\)  and \(l_4\in L(p_4)\) ) that are incident to \(q\) . The converse, however is not true; three lines \(l_1,l_2,l_4\)  intersect in a rectangle of dimensions roughly \(\delta \times \delta /d\) ; the long axis of this rectangle makes an angle \(\lesssim d^{-1}\)  with the long axis of \(R\)  (since both of these rectangles have eccentricity about \(d^{-1}\) , the two rectangles are as “parallel” as possible). By (REF ), there could be as many as \(\lessapprox d^{-\alpha }V\)  points \(q\in Q_1\)  in this rectangle that are incident to \(l_1,l_2\)  and \(l_4\) . Let \(Q_2\subset Q_1\) , with
\(\#Q_2\gtrapprox d^{\alpha }V^{-1}(\#Q_1)\gtrapprox d^{-\alpha }\delta ^{-10\alpha }(\#L)^4(\#P)^{-8}V^5 M^{-1},\) 
so that each rectangle dimensions \(\delta \times \delta /d\)  that is parallel to \(R\)  contains at most one point from \(Q_2\) .
Let \(\tilde{\delta }= \delta /d\) . Let \(\tilde{T}\colon \mathbb {R}^2\rightarrow \mathbb {R}^2\)  be the composition of a translation, rotation, and dilation that first sends \(p_1\)  to \((0,0)\) , then sends \(p_4\)  to a point on the \(x\) -axis, and finally stretches \(\mathbb {R}^2\)  in the \(x\) -direction, so that the image of \(p_4\)  becomes \((1,0)\) . The image of \(R\)  under \(\tilde{T}\)  is a rectangle of dimensions \(\approx 1 \times 1\) . Let \(\tilde{p}_i\)  and \(\tilde{L}(p_i)\) , \(i=1,2,3,5\)  be the image of \(p_i\)  and \(L(p_i)\) , respectively, under \(\tilde{T}\) . Let \(\tilde{Q}_2\)  be the image of \(Q_2\)  under \(T\)
We have that \(\tilde{p}_1 = (0,0)\) , \(\tilde{p}_4=(1,0)\) , and \(\tilde{p}_2\)  is contained in the \(\delta \) -neighborhood of the \(x\) -axis and has distance \(\approx 1\)  from \(\tilde{p}_1\)  and \(\tilde{p}_4\) . Abusing notation, we will replace \(\tilde{p}_2\)  with a point on the \(x\) -axis, and modify the set \(\tilde{L}(p_2)\)  accordingly.
The purpose of this transformation is that now the points in \(\tilde{Q}_2\)  are \(\tilde{\delta }\) -separated, and each \(q\in \tilde{Q}_2\)  satisfies
\(\operatorname{dist}(q, x\textrm {-axis})\approx 1,\quad \operatorname{dist}(q, \overline{\tilde{p}_1\tilde{p}_5})\approx 1,\quad \operatorname{dist}(q, \overline{\tilde{p}_4\tilde{p}_5})\approx 1.\) 
Finally, the lines in \(\tilde{L}(p_1)\)  satisfy a re-scaled version of the non-concentration estimate (REF ). For each unit vector \(v\in S^1\)  and each \(r\ge \delta \) , we have
\(\# \lbrace  l \in \tilde{L}(p_1), \angle (l, v) \le r \rbrace  \lessapprox (dr/\delta )^\alpha =(r/\tilde{\delta })^{\alpha },\) 
and similarly for \(\tilde{L}(p_2),\)  \(\tilde{L}(p_4),\)  \(\tilde{L}(p_5).\)

Applying a projective transformation.
Next, we apply a projective transformation \(T^\dag \)  that sends the \(x\) -axis to the line at infinity. We choose this transformation so that lines containing \((0,0)\)  (and in particular, lines in \(\tilde{L}(p_1)\) ) are mapped to vertical lines; lines containing \((1,0)\)  (and in particular, lines in \(\tilde{L}(p_4)\) ) are mapped to horizontal lines; and lines containing \(\tilde{p}_3\)  (and in particular, lines in \(\tilde{L}(p_3)\) ) are mapped to lines with slope 1.
Recall that \(\tilde{p}_1,\)  \(\tilde{p}_2\) , and \(\tilde{p}_3\)  have separation \(\approx 1\) ; \(\tilde{Q}_2\)  is contained in a ball centered at the origin of radius \(\approx 1\) ; and
by (REF ), \(\operatorname{dist}(\tilde{Q}_2,\ x\textrm {-axis})\approx 1\) . Thus \(T^\dag \)  does minimal damage to \(\tilde{Q}_2\) —the set remains contained in a ball centered at the origin of radius \(\approx 1\) ; it is \(\approx \tilde{\delta }\)  separated; and if \(l\in \tilde{L}(p_1)\)  was \(\tilde{\delta }\) -incident to some \(q\in \tilde{Q}_2\) , then the image of \(l\)  remains \(\approx \tilde{\delta }\) -incident to the image of \(q\) .
Recall from (REF ) that \(\operatorname{dist}(p_5, \overline{p_1p_4})\approx 1\) , and thus \(\operatorname{dist}(\tilde{p}_5, x\textrm {-axis})\approx 1\) . This means that the image of \(\tilde{p}_3\)  under \(T^\dag \)  has distance \(\lessapprox 1\)  from the origin, so we can apply a translation and suppose that the image of \(\tilde{p}_3\)  is \((0,0)\) . After this translation we still have that the image of \(\tilde{Q}_2\)  is contained in a ball centered at the origin of radius \(\approx 1\) . Abusing notation, we will continue to refer to this transformation as \(T^\dag \)
Let \(L(p_1)^\dag \) , \(L(p_2)^\dag \) , \(L(p_4)^\dag \) , \(L(p_5)^\dag \)  denote the images of \(\tilde{L}(p_1)\) , \(\tilde{L}(p_2)\) , \(\tilde{L}(p_4)\) , \(\tilde{L}(p_5)\)  under \(T^\dag \) . \(T^\dag \)  maps the line \(\overline{\tilde{p}_1,\tilde{p}_5}\)  to the \(y\) -axis, and the line \(\overline{\tilde{p}_4,\tilde{p}_5}\)  to the \(x\) -axis. Thus by (REF ), each \(q^\dag \in T^\dag (\tilde{Q}_2)\)  has distance \(\approx 1\)  from the \(x\)  and \(y\)  axes. For convenience, we will suppose that each \(q^\dag \in T^\dag (\tilde{Q}_2)\)  is contained in the first quadrant; this can always be achieved by applying a suitable reflection and refining \(\tilde{Q}_2\)  by a factor of at most 4. Next, by dyadic pigeonholing, we can suppose there are numbers \(M_x,M_y\approx 1\)  so that after refining \(\tilde{Q}_2\)  by at most \(|\log \delta |^{-1}\) , we have that \(T^\dag (\tilde{Q}_2)\)  is contained in \([M_x, 2M_x]\times [M_y, 2M_y]\) . Thus if we compose \(T^\dag \)  with the anisotropic scaling \(S(x,y)=(M_x^{-1}x, M_y^{-1}y)\)  and define \(Q^\dag = S\circ T^\dag (\tilde{Q} \cap [M_x, 2M_x]\times [M_y, 2M_y])\) , then \(Q^\dag \subset [1,2]^2\) . Abusing notation slightly, we will replace \(L(p_1)^\dag \) , \(L(p_2)^\dag \) , \(L(p_4)^\dag \) , \(L(p_5)^\dag \)  with their image under \(S\) .
Finally, we will choose a value of \(\delta ^\dag \approx \tilde{\delta }\)  so that after further refining \(Q^\dag \)  by a \(\approx 1\)  factor we, we can ensure that \(Q^\dag \)  is \(\delta ^\dag \) -separated, and if \(l\in \tilde{L}(p_1)\)  was \(\tilde{\delta }\) -incident to some \(q\in \mathcal {Q}_2\) , then the image of \(l\)  will be \(\delta ^\dag \) -incident to the image of \(q.\)
Recall that the lines in \(L(p_1)^\dag \)  are parallel to the \(y\) -axis; let \(X\subset \mathbb {R}\)  be the set of \(x\) -intercepts of these lines. The lines in \(L(p_4)^\dag \)  are parallel to the \(x\) -axis; let \(Y\subset \mathbb {R}\)  be the set of \(y\) -intercepts of these lines. The lines in \(L(p_2)^\dag \)  are of the form \(x+y = t\) ; let \(Z\subset \mathbb {R}^2\)  be the set of \(t\) -values for such lines. The lines in \(L(p_5)\dag \)  are of the form \(y = mx\) ; let \(W\)  be the set of slopes (i.e. \(m\)  values) of these lines.
The sets \(X, Y, Z, W\)  are \(\approx \delta ^\dag \)  separated. Since each \(q^\dag \in Q_2^\dag \)  is contained in \([1,2]^2\) , each line \(l\in L(p_1)^\dag \)  that is incident to at least one point of \(Q_2^\dag \)  must have \(x\) -intercept in \([1,2]\) . Similarly, each line \(l\in L(p_4)^\dag \)  that is incident to at least one point of \(Q_2^\dag \)  must have \(y\) -intercept in \([1,2]\) . Thus we can suppose that \(X,Y\subset [1,2]\)  (i.e. we can discard any elements of \(X\)  or \(Y\)  for which this condition fails, since such lines cannot be incident to any points in \(Q_2^\dag \) ).
The angular non-concentration (REF ) becomes a spatial non-concentration condition on the sets \(X\)  and \(Y\) . Specifically, for each interval \(J\subset \mathbb {R}\)  we have
\(\begin{split}&\#(X \cap J) \lessapprox (|J|/\delta ^\dag )^{\alpha },\\&\#(Y \cap J) \lessapprox (|J|/\delta ^\dag )^{\alpha }.\end{split}\) 
Note that the “\(\lessapprox \) ” in the above inequalities represents a multiplicative factor of the form \(\delta ^{-C\varepsilon }\) , not \((\delta ^\dag )^{-C\varepsilon }\) .
(REF ) becomes
\(\begin{split}&\#X\lessapprox \delta ^{-\alpha }(\#L)(\#P)^{-1}V M^{-1},\\&\#Y\lessapprox \delta ^{-\alpha }(\#L)(\#P)^{-1}V M^{-1},\\&\#Z\lessapprox \delta ^{-\alpha }(\#L)(\#P)^{-1}V M^{-1},\\&\#W\lessapprox \delta ^{-\alpha }(\#L)(\#P)^{-1}V M^{-1}.\end{split}\) 
Recall that \(X_{\delta ^\dag } = N_{\delta ^\dag }(X)\cap (\delta ^\dag \mathbb {Z})\) . Let \(E\subset X_{\delta ^\dag }\times Y_{\delta ^\dag }\) , with \((x,y)\in E\)  if there are points \(x^{\prime }\in X\) , \(y^{\prime }\in Y\)  with \(|x-x^{\prime }|\le \delta ^\dag ,\)  \(|y-y^{\prime }|\le \delta ^\dag ,\)  so that the corresponding lines \(l_1\in L(p_1)^\dag \) , \(l_4\in L(p_4)^\dag \)  with intercepts \(x^{\prime }\)  and \(y^{\prime }\)  are \(\delta ^\dag \) -incident to a common point of \(Q_2^\dag \) . By (REF ),
\(\# E\gtrapprox d^{-\alpha }\delta ^{-10\alpha }(\#L)^4(\#P)^{-8}V^5 M^{-1}.\) 
Furthermore, if \((x,y)\in E\) , then the corresponding point \(q\in \tilde{Q}_2^\dag \)  is incident to a line of the form \(x+y=t\) , where \(|t-(x-y)|\le 4\delta ^\dag \) . This means that \(x-y\)  is an element of \(\delta ^\dag \mathbb {Z}\)  that is contained in \(N_{4\delta ^\dag }(Z)\) . But since \(Z\)  is \(\approx \delta ^\dag \)  separated, by (REF ) we have that
\(\#(X_{\delta ^\dag } \overset{E}{-} Y_{\delta ^\dag })\le 4\mathcal {E}_{\delta ^\dag }(Z)\lessapprox \#Z \lessapprox \delta ^{-\alpha }(\#L)(\#P)^{-1}V M^{-1}.\) 

Bounding the sum-set.
Next, we will use the Balog-Szemerédi-Gowers theorem to find a large set \(A_0\subset X_{\delta ^\dag }\)  whose sum-set is small; use Lemma REF  to extract a set \(A_0\subset X_{\delta ^\dag }\)  with
\(\begin{split}\# A_0 &\gtrapprox \frac{d^{-\alpha }\delta ^{-10\alpha }(\#L)^4(\#P)^{-8}V^5 M^{-1}}{\delta ^{-\alpha }(\#L)(\#P)^{-1}V M^{-1}}\\&= \Big (d^{-2\alpha }\delta ^{-8\alpha }(\#L)^3(\#P)^{-7}V^4\Big )(\delta ^\dag )^{-\alpha },\end{split}\) 
and
\(\begin{split}\#(A_0-A_0) &\lessapprox \frac{\big (\delta ^{-\alpha }(\#L)(\#P)^{-1}V M^{-1}\big )^{12}}{\big (d^{-\alpha }\delta ^{-10\alpha }(\#L)^4(\#P)^{-8}V^5 M^{-1} \big )^6} (\#A)\\&= \Big (d^{6\alpha }\delta ^{48\alpha }(\#L)^{-12}(\#P)^{36}V^{-18}M^{-6}\Big ) (\#A).\end{split}\) 
Next, we will find a large set \(A\subset A_0\) , so that \(A+A\)  is has small covering number at every scale (c.f. Item REF  from Proposition REF ). Our arguments are similar to the proof of Lemma 2.1 from [2]}. Let \(k=1/\varepsilon \)  and select \(T\)  so that \(2^{kT}\le \delta ^\dag \le 2^{k(T+1)}\) . Apply Lemma REF  to \(A_0\) ; let \(A\subset A_0\)  and \((N_0,\ldots ,N_{k-1})\)  be the output from that lemma. \(A\)  continues to satisfy (REF ). We will show that for each \(\delta ^\dag \le \rho \le 1,\)  we have
\(\mathcal {E}_{\rho }(A-A)\lessapprox \Big (d^{6\alpha }\delta ^{48\alpha }(\#L)^{-12}(\#P)^{36}V^{-18}M^{-6}\Big ) \mathcal {E}_{\rho }(A).\) 
First, we will verify (REF ) for \(\rho \)  of the form \(2^{-jT},\ 1\le j\le k\) . Since each \(2^T\) -adic interval of length \(2^{-jT}\)  contains \(\prod _{i=j}^{k-1}N_i\)  points from \(A\) , we have
\(\#(A-A) = \mathcal {E}_{\delta }(A-A)\ge \prod _{i=j}^{k-1}N_i\mathcal {E}_{\rho }(A-A).\) 
Thus
\(\mathcal {E}_{\rho }(A-A) \le \Big (\prod _{i=j}^{k-1}N_i\Big )^{-1}\#(A-A)\lessapprox \Big (\prod _{i=j}^{k-1}N_i\Big )^{-1} \Big (d^{6\alpha }\delta ^{48\alpha }(\#L)^{-12}(\#P)^{36}V^{-18}M^{-6}\Big ) (\#A_0).\) 
(REF ) now follows from the inequality \(\#A\approx \#A_0\) , and the observation that \(\mathcal {E}_{\rho }(A)=\big (\prod _{i=j}^{k-1}N_i\big )^{-1}(\#A)\) .
Next, to verify (REF ) for arbitrary \(\rho ,\)  select \(1\le j\le k\)  with \(2^{-jT}\le \rho \le 2^{-(j-1)T}\)  and let \(\rho ^{\prime }=2^{-jT}\) . Then (REF ) follows from the observation that \(\mathcal {E}_{\rho }(A-A) \le \mathcal {E}_{\rho ^{\prime }}(A-A),\)  and \(\mathcal {E}_{\rho ^{\prime }}(A)\approx \mathcal {E}_{\rho }(A)\) .

Bounding the multiplicative energy.
We still have that every point in \(E \cap (A\times Y_{\delta ^\dag })\)  is \(\delta ^\dag \) -incident to a line of the form \(y = mx\) , with \(m\in W\) . Thus by Cauchy-Schwarz,
\(\#\lbrace (p,p^{\prime })\in E \cap (A\times Y)\colon p,p^{\prime }\ \textrm {are \) 
†\(-incident to a common line}\ y=mx,\ m\in W \rbrace  \gtrapprox \frac{\big (\#(E \cap (A\times Y))\big )^2}{\#W}.\)\(Since \) A,Y[1,2]\(, each pair \) (p,p')\( in the above set contributes to the multiplicative energy \) E†(A,Y)\(, and thus we have\)\(E^{\delta ^\dag }_{\times }(A,Y)\gtrapprox \frac{\big (\#(E \cap (A\times Y))\big )^2}{\#W}.\)\(Since \) Y[1,2]\( is \) †\(-separated, we have \) E†(Y,Y)(#Y)3\(. By Lemma \ref {energyCauchySchwartz}, we have\)\(E^{\delta ^\dag }_{\times }(A,A)\ge \frac{\big (E^{\delta ^\dag }_{\times }(A,Y)\big )^2}{E^{\delta ^\dag }_{\times }(Y,Y)}\gtrapprox \frac{\big (\#(E \cap (A\times Y))\big )^4}{(\#W)^2(\#Y)^3}.\)\(But\)\(\#\big (E \cap (A\times Y)\big )\gtrapprox (\# E)\frac{\# A}{\# X},\)\(so we get\begin{equation}\begin{split}E^{\delta ^\dag }_{\times }(A,A) &\gtrapprox \frac{(\#E)^4(\#A)^4}{(\#W)^2(\#X)^4(\#Y)^3}\\&\gtrapprox \frac{(\#E)^5}{(\#W)^2(\#X)^4(\#Y)^4} (\#A)^3\\&\gtrapprox \big (d^{-5\alpha } \delta ^{-40\alpha }(\#L)^{10}(\#P)^{-30}V^{15} M^5\big )(\#A)^3.\end{split}\end{equation}\)

Applying discretized sum-product.
To summarize, we have a \(A\subset [1,2]\cap (\delta ^\dag \mathbb {Z})\) , with
\(\# A \gtrapprox \Big (d^{2\alpha }\delta ^{8\alpha }(\#L)^{-3}(\#P)^{7}V^{-4}\Big )^{-1} (\delta ^\dag )^{-\alpha },\) 
which satisfies the non-concentration condition
\(\#(A\cap J)\lessapprox |J|^{\alpha } (\delta ^\dag )^{-\alpha },\) 
and
\(\begin{split}\mathcal {E}_{\rho }(A-A)&\lessapprox \Big (d^{6\alpha }\delta ^{48\alpha }(\#L)^{-12}(\#P)^{36}V^{-18}M^{-6}\Big ) \mathcal {E}_{\rho }(A),\quad \delta ^\dag \le \rho \le 1,\\E^{\delta ^\dag }_{\times }(A,A) & \gtrapprox \Big (d^{5\alpha } \delta ^{40\alpha }(\#L)^{-10}(\#P)^{30}V^{-15} M^{-5}\Big )^{-1}(\#A)^3.\end{split}\) 
Comparing this with Proposition REF  and using the bounds (REF ), (REF ), and (REF ), we conclude that
\(\begin{split}& \Big (d^{2\alpha }\delta ^{8\alpha }(\#L)^{-3}(\#P)^{7}V^{-4}\Big )^6\\&\cdot \Big (d^{6\alpha }\delta ^{48\alpha }(\#L)^{-12}(\#P)^{36}V^{-18}M^{-6}\Big )^{18+8\alpha }\Big (d^{5\alpha } \delta ^{40\alpha }(\#L)^{-10}(\#P)^{30}V^{-15} M^{-5}\Big )^{8+4\alpha } \\&= d^{4\alpha (40+17\alpha )} \delta ^{16\alpha (77+34\alpha )} (\#L)^{-2(157+68\alpha )} (\#P)^{6(155+68\alpha )} V^{-12(39+17\alpha )} M^{-4(37+17\alpha )} \\&= (\delta ^{2\alpha } \#P)^{6(155+68\alpha )} (\delta ^{2\alpha } \#L)^{-2(157+68\alpha )} d^{4\alpha (40+17\alpha )} V^{-12(39+17\alpha )} M^{-4(37+17\alpha )}\\&\hspace{284.52756pt} \gtrapprox (\delta ^\dag )^{-\alpha (1-\alpha )},\end{split}\) 
i.e.
\((\delta ^{2\alpha } \#P)^{6(155+68\alpha )} \gtrapprox d^{-3\alpha (53+23\alpha )} M^{4(37+17\alpha )} \delta ^{-\alpha (1-\alpha )}(\delta ^{2\alpha }\#L)^{2(157+68\alpha )}V^{12(39+17\alpha )}.\) 
It is now clear that the worst case occurs when \(d\)  is as large as possible (i.e. \(d\sim 1\) ) and \(M\)  is as small as possible (i.e. \(M\sim 1\) ). We conclude that
\(\#P \gtrapprox \delta ^{-2\alpha -c(\alpha )}(\delta ^{2\alpha }\#L)^{\frac{157+68\alpha }{3(155+68\alpha )}}V^{\frac{2(39+17\alpha )}{155+68\alpha }},\quad c(\alpha ) = \frac{\alpha (1-\alpha )}{6(155+68\alpha )}.\) 
The final step is to verify that for all \(\alpha \in (0,1)\) , \(\frac{157+68\alpha }{3(155+68\alpha )}\le 1\)  and \(\frac{2(39+17\alpha )}{155+68\alpha }\le c(\alpha )/\alpha \) .
Remark 4.4 When \(c=1/2\) , the discretized sum-product theorem (Proposition REF ) gives a gain of roughly \(\delta ^{-1/100}\) . The final gain Proposition REF  is roughly 50 times, or two orders of magnitude, worse than this. Roughly speaking, one order of magnitude comes from repeated pigeonholing and use of Cauchy-Schwarz, while the other order of magnitude comes from the use of Balog-Szemerédi-Gowers. While some further optimizations are likely possible, it appears that such optimizations would yield rather modest gains. In particular, Proposition REF  requires that the difference set \(A-A\)  be small (rather than merely requiring that \(E_{+}(A,A)\)  be large), so it appears difficult to avoid using the Balog-Szemerédi-Gowers theorem). While a more efficient sequence of pigeonholing and Cauchy-Schwarz arguments might improve the final bound, since the current argument only loses (roughly) one order of magnitude at this step, any improvement here would at best yield a modest improvement in the final bound.


From Proposition REF  to discretized Furstenberg
Our goal in this section is to prove that Proposition REF  implies Proposition REF , and hence Theorem REF . First, we will briefly describe the differences between Propositions REF  and REF . Both are incidence theorems about collections of points and lines, but Proposition REF  requires two additional non-concentration conditions.
Property REF  is sometimes called “robust transversality,” and it asserts that for each \(p\in P\) , a typical pair of lines incident to \(p\)  point in well-separated directions. With this condition, Proposition REF  can also be viewed as a bilinear incidence theorem. Thus our reduction from Proposition REF  to REF  will employ “bilinear to linear” type arguments, which were first used by Tao, Vargas, and Vega in the context of the Kakeya and Restriction problems [3]}. A prototype for the specific arguments used here can be found in [4]}. See also [1]} for a similar argument in a slightly different setting.
Property (REF ) is also an assertion about the distribution of directions \(\lbrace \operatorname{dir}(l)\colon (p,l)\in I\rbrace \) . This property imposes stronger non-concentration conditions at small scales (specifically at length scales between \(\delta \)  and \((\delta ^{-2\alpha }/\#P)^{1/\alpha }\) ), but it says nothing about non-concentration at larger scales. To the best of our knowledge, the arguments used to obtain the assertion in Property (REF ) are novel, and we hope that similar arguments might be helpful for related problems.
[Proof of Proposition REF ]
Fix \(0<\alpha <1\)  and \(c<c(\alpha )=\frac{\alpha (1-\alpha )}{6(155+68\alpha )}\) . We will show that if the parameters \(\varepsilon >0\)  and \(C\)  are chosen appropriately, then for all \(\delta >0\)  sufficiently small, if \(E\supset \bigcup _{l\in L_0}P_l\)  is a discretized \((\alpha ,2\alpha )\) -Furstenberg set (with implicit parameters \(\varepsilon \)  and \(C\) ), then
\(\mathcal {E}_\delta (E) \ge \delta ^{-2\alpha -c}.\) 
In particular, (REF ) will follow from the estimate
\(\mathcal {E}_\delta \Big ( \bigcup _{l\in L_0}P_l \Big ) \gtrapprox \delta ^{-2\alpha -c(\alpha )}.\) 
Constructing the incidence arrangement.
Our goal is to construct a \(\delta \) -separated set \(P\subset \bigcup _{l\in L_0}P_l\) , and an incidence relation \(I\subset P\times L\)  that satisfies the hypotheses of Proposition REF . While it is tempting to simply choose \(P\)  to be a maximal \(\delta \) -separated subset of \(\bigcup _{l\in L_0}P_l\) , it is possible that a set constructed in this fashion will have the following undesirable property: let \(R\)  be a rectangle of dimensions \(r\times \delta \) , so that many lines \(l\in L_0\)  intersect \(R\)  in a line-segment of length roughly \(r\) ; while Item (REF ) from Definition REF  guarantees that each set \(R\cap P_l\)  has cardinality \(\lessapprox (r/\delta )^\alpha \) , the union \(R\cap \bigcup P_l\)  might be much larger. For technical reasons that will become apparent later in the proof, we would like to avoid this type of incidence arrangement.
To avoid this problem, we will use a stopping-time type argument to construct an incidence arrangement \((P,L,I)\)  so that for each rectangle \(R\)  of this type, only a few lines from \(L_0\)  that have large intersection with \(R\)  contribute points to \(R\) .
Let \(k = \lfloor 1/\varepsilon \rfloor \)  and select \(T\)  so that \(2^{-kT}\le \delta /6< 2^{-k(T-1)}\) . Refining each set \(\lbrace P_l\rbrace _{l\in L_0}\)  by a factor of at most \((2\delta )^{-\varepsilon }\) , we can suppose that each set \(P_l\)  is discretized at scale \(2^{-kT}\) . Apply Corollary REF  to the collection of sets \(\lbrace P_l\rbrace _{l\in L_0}\) , and let \(L_1\subset L_0\) ; \(\lbrace P_l^{\prime }\rbrace _{l\in L_1};\)  \((N_0,\ldots ,N_{k-1})\)  be the output from the corollary. We have
\(\sum _{l\in L_1}\#P_l^{\prime } \gtrapprox \sum _{l\in L}\#P_l \gtrapprox \delta ^{-3\alpha }.\) 
Enumerate the elements of \(L_1\)  as \(l_1,\ldots ,l_M\) , \(M\approx \delta ^{-2\alpha }\) . Will will construct a point set \(P_1\subset \bigcup _{l\in L}P_l\)  and an incidence relation \(I_1\subset P_1\times L_1\)  as follows.
We begin by setting \(P_1=\emptyset \)  and \(I_1 = \emptyset \) . Let \(m=1\) , and add all of the points in \(P_{l_1}^{\prime }\)  to \(P_1\) ; add all of the incidences \(\lbrace  (p,l_1)\colon p\in P_{l_1}^{\prime }\rbrace \)  to \(I_1\) . For each \(2^T\) -adic square \(S\)  for which \(P_{l_1}^{\prime }\cap S\ne \emptyset \) , we say that \(l_1\)  has contributed points in square \(S\). We have now “processed” the line \(l_1\) .
Suppose that we have processed the lines \(l_1,\ldots ,l_{m-1}\) . We shall process the line \(l_m\)  as follows. To begin, we declare all \(2^T\) -adic squares inside \([0,1]^2\)  to be “undominated.” We perform the following procedure for each \(j=0,\ldots , k\) : consider each \(2^T\) -adic square \(S\subset [0,1)^2\)  of side-length \(2^{-jT}\)  that is not contained in a (larger) dominated square.

If \(S\cap {P^{\prime }_{l_m}}=\emptyset \) , we ignore \(S\) , and it remains undominated (squares of this type will not be relevant for successive steps of the algorithm).

If instead \(P^{\prime }_{l_m}\cap S\ne \emptyset \) , and hence \(\#(P^{\prime }_{l_m}\cap S)=\prod _{i=j}^{k-1}N_i\) , then either (A) or (B) must hold:
(A):
there is at least one line \(l_n,\ n<m\)  so that:
(i):
\(l_n\)  has contributed points in \(S\) ,
(ii):
\(\angle (l_m, l_n)\le (\frac{\pi }{2})2^{(j-k)T}\) ,
(iii):
\(\operatorname{dist}(l_m\cap S, l_n\cap S)\le (\sqrt{2})2^{-kT}.\)

(B):
no such line exists.

If Option (A) occurs, select a line \(l_n\)  with this property; add the pairs \(\lbrace (p,l_m)\colon p\in S\cap P^{\prime }_{l_n}\rbrace \)  to \(I_1\) ; and mark the square \(S\)  as dominated. We will say that “\(l_m\)  is dominated by \(l_n\)  on square \(S\) .” Note that the conditions in Option (A) imply that \(l_n\cap S \subset N_{\delta }(l_m)\)  (indeed, there is a point in \(S\)  where \(l_m\)  and \(l_n\)  have separation at most \((\sqrt{2}) 2^{kT}\le \delta /4\) , and the condition on \(\angle (l_m,l_n)\)  then ensures that \(l_n\cap S \subset N_{\delta }(l_m)\) . Thus each of the incidences \((p,l_m)\)  added in this step satisfy \(p\in N_{\delta }(l_m)\) . If Option (B) occurs, the square \(S\)  remains undominated.

Repeat this process for \(j=1,2,\ldots ,k\) . We say a \(2^T\) -adic square \(S\)  is a maximal purely undominated square if \(S\)  does not contain a dominated square, but every \(2^T\) -adic square \(S^{\prime }\supset S\)  does contain a dominated square. Each undominated square \(S\)  is contained in a unique maximal purely undominated square. For each maximal purely undominated square \(S\)  for which \(S\cap P_{l_m}^{\prime }\ne \emptyset \) , add the points in \(S\cap P_{l_m}^{\prime }\)  to \(P_1\) , and add the corresponding incidences \((p,l_m)\)  to \(I_1\) . For each \(2^T\) -adic square \(S\)  (maximal or not) for which \(P_{l_m}^{\prime }\cap S\)  is undominated, we say that \(l_m\)  has contributed points in square \(S\) .
After the above steps have been completed, we have now processed the line \(l_m\) . Observe that at this point, the incidence arrangement \((P_1,L_1,I_1)\)  has the following properties:

If \((p,l_m)\in I_1\) , then \(p\in N_{\delta }(l_m)\)

For each \(2^T\) -adic square \(S\) , we have
\(\#\lbrace p\in P_1\cap S\colon (p, l_m)\in I_1 \rbrace =\#(P_{l_m}^{\prime }\cap S).\) 

As a consequence of (REF ) when \(S=[0,1)^2\) , we have \(\# I_1(l_m)\gtrapprox \delta ^{-\alpha }.\)  Furthermore, if \(\delta \le r\le 1\) , then
\(\#(B(x,r)\cap I_1(l_m))\lessapprox (r/\delta )^{\alpha }\) 
for all balls \(B(x,r)\) . Indeed, choose \(j\)  so that \(2^{-jT}\le r< 2^{-(j-1)T}\) . Then \(B(x,r)\)  intersects \(\approx 1\)  squares of side-length \(2^{-jT}\) , so (REF ) follows from (REF ), plus the fact that \(P_{l_m}^{\prime }\)  is a \((\delta ,\alpha )_1\) -set.
As additional lines are processed, we will add more points to \(P_1\)  and more incidences to \(I_1\) , but this will not disrupt the properties of \(l_m\)  described above.
After all of the lines in \(L_1\)  have been processed in the above fashion, the incidence arrangement \((P_1, L_1, I_1)\)  satisfies Properties (B) (with \(V=1\) ) and (C) from Proposition REF , as well as the estimate \(\#I_3\approx \delta ^{-3\alpha }\) . We claim that \(P_1\)  is discretized at scale \(2^{-kT}\) , and thus \(\lessapprox 1\)  points \(p\in P_1\)  can be contained in any ball of radius \(\delta \) . To prove this claim, note that if \(S\)  is a square of side-length \(2^{-kT}\) , then at most one line \(l\in L_1\)  can contribute points to \(S\) —if two such lines \(l,l^{\prime }\)  both contributed points to \(S\) , then we must have that either \(\angle (l,l^{\prime })> \pi /2\)  (which is impossible), or \(\operatorname{dist}(l\cap S, l^{\prime }\cap S)> (\sqrt{2})2^{-kT}=\operatorname{diameter}(S)\)  (which is impossible).
Since \(P_1\)  is discretized at scale \(2^{-kT}\) , in order to establish (REF ) it suffices to prove that
\(\#P_1 \gtrapprox \delta ^{-2\alpha -c(\alpha )}.\) 
Finally, we observe that if \(p\in P_1\) , if \(1\le j\le k-1\) , and if \(S\)  is a square of side-length \(2^{-jT}\)  that contains \(p\) , then there is exactly one line \(l\in I_1(p)\)  that contributes points in square \(S\) .

Non-concentration of lines through a point.
Our next task is to extract a set of points \(P_2\subset P_1\)  and a set of incidences \(I_2\subset I_1\)  so that for each \(p\in P_2\) , the distribution of directions of lines incident to \(p_2\)  satisfies Property (F).
For each \(p\in P_1\) , we will identify the set \(I_1(p)\)  with a subset of \([0,1]\)  by identifying the line \(y=mx+b\)  with the point \(m\) ; call this set \(A_p\) . Since \(L_0\)  is \(\delta \) -separated, at most 16 points in \(A_p\)  can intersect any interval of length \(2^{-kT}\le \delta /6\)  (indeed, the set of such points in \(A_p\)  corresponds to lines in \(L_1\)  whose image under \(\iota \)  is contained in a ball of radius \(2\delta \) ). In particular, by refining each set \(A_p\)  by a factor of 16, we can suppose that \(A_p\)  is \(2^{-kT}\) -discretized. By removing on additional point if necessary, we can also suppose that \(A_p\subset [0,1)\) .
Apply Corollary REF  to the collection of sets \(\lbrace A_p\rbrace _{p\in P_1}\) , and let \(P_2\subset P_1\) ; \(\lbrace A_p^{\prime }\rbrace _{p\in P_2};\)  \((M_0,\ldots ,M_{k-1})\)  be the output from the corollary. Each set \(A_p^{\prime }\)  has cardinality \(\prod _{j=0}^{k-1}M_j\) , and
\((\#P_2)\prod _{j=0}^{k-1}M_j = \sum _{p\in P_2}\#A_p^{\prime } \approx \sum _{p\in P_1}\#A_p^{\prime }\approx \#I_1\approx \delta ^{-3\alpha }.\) 
Define \(I_2\)  to consist of those incidences \((p,l)\in P_2\times L_1\)  so that the slope of \(l\)  is in \(A_p^{\prime }.\)
Recall that if \(0\le j\le k\)  and if \(J\subset [0,1)\)  is an \(2^T\) -adic interval of side-length \(2^{-jT}\)  that meets \(A_p\) , then \(\#(J\cap A_p^{\prime })= \prod _{i=j}^{k-1}M_i\) . Our next task is to control the size of this product. In theory, each \(M_i\)  could be as large as \(2^{T}\) . We will show that on average, each \(M_i\)  has size at most \(2^{\alpha T}\) . More precisely, we will prove that there exists a constant \(C_0=C_0(\alpha )\)  so that
\(\prod _{i=j}^{k-1}M_i \le \delta ^{-C_0\varepsilon } \Big ( \frac{2^{(j-k)T}}{\delta }\Big )^\alpha , \quad \textrm {for each}\ j = 0,\ldots ,k-1.\) 
We will prove (REF ) by induction—the base case will be when \(j\)  is large (i.e. \(j\)  is close to \(k-1\) , which corresponds to small length scales), and the induction step will go from \(j\)  to \(j-1\) . Observe that since the points in \(A_p^{\prime }\)  are \(2^{-kT}\) -separated, we have the trivial estimate \(M_{i} \le 2^{T}\le \delta ^{-2\varepsilon }.\) 
Thus since \(2^{(j-k)T}\delta ^{-1}\ge 2^{-kT}\delta ^{-1}\ge \delta ^{2\varepsilon },\)  we have
\(\prod _{i=j}^{k-1}M_i\le \delta ^{-(2k-2j-2)\varepsilon }\Big ( \frac{2^{(j-k)T}}{\delta }\Big )^\alpha ,\) 
and thus (REF ) holds for \(j \ge k - (C_0-2)/2\) .
Suppose now that (REF ) has been established for all \(j^{\prime }>j\) ; our task is to prove (REF ) for \(j\) . If \(M_j\le 2\)  then this follows immediately from (REF ) for \(j+1\) . Suppose now that \(M_j\ge 3\) .
Since \((\#I_2)(\#L_1)\gtrapprox \delta ^{-\alpha }\) , there exists a line \(l\in L\)  with \(\# I_2(l)\gtrapprox \delta ^{-\alpha }\) . Fix this choice of \(l\) . Define \(\rho = 2^{(j-k)T}\) . By (REF ), each ball \(B\)  of radius \(\delta ^{1-\varepsilon }\rho ^{-1}\)  satisfies
\(\# ( I_2(l) \cap B)\lessapprox (\delta ^{1-\varepsilon }\rho ^{-1}/\delta )^\alpha \lessapprox \rho ^{-\alpha },\) 
and hence we can select a set \(P^{\prime }\subset I_2(l)\)  that is \((\delta ^{1-\varepsilon }\rho ^{-1})\) -separated, with \(\#P^{\prime }\gtrapprox (\delta /\rho )^{-\alpha }\) .
Let \(m\)  be the slope of \(l\) . For each \(p\in P^{\prime }\) , we have that \(m\in A_p\) , and hence the \(2^T\) -adic interval \(J\)  of length \(\rho \)  containing \(m\)  intersects \(A_p\) . Thus for each \(p\in P^{\prime },\)  there are \(\ge \prod _{i=j}^{k-1}M_i \)  lines \(l^{\prime } \in I_2(p)\)  with \(|m-m^{\prime }|\le \rho \)  (here \(m^{\prime }\)  is the slope of \(l^{\prime }\) ). On the other hand, the lines \(l^{\prime } \in I_2(p)\)  with \(|m-m^{\prime }|\le 2^{-T}\rho \)  are contained in a union of at most two \(2^T\) -adic intervals of length \(2^{-(j-1)T}\) ; each of these intervals contains \(\prod _{i=j+1}^{k-1}M_i = M_j^{-1}\prod _{i=j}^{k-1}M_i\le \frac{1}{3}\prod _{i=j}^{k-1}M_i\)  points.
Thus for each \(p\in P^{\prime }\)  there are at most \(\frac{2}{3}\prod _{i=j}^{k-1}M_i\)  lines \(l^{\prime }\in I_2(p)\)  with \(|m-m^{\prime }|\le 2^{-T}\rho \) . We conclude that
\(\# \lbrace l^{\prime } \in I_2(p)\colon \delta ^{\varepsilon }\rho \le 2^{-T}\rho \le |m-m^{\prime }|\le \rho \rbrace  \ge \frac{1}{3}\prod _{i=j}^{k-1}M_i.\) 
Call the above set of lines \(L(p)\) . Since the points in \(P^{\prime }\)  are \(\delta ^{1-\varepsilon }\rho ^{-1}\)  separated and contained in the \(\delta \) -neighborhood of \(l\) , if \(p,p^{\prime }\in P^{\prime }\)  are distinct, then the sets \(L(p)\)  and \(L(p^{\prime })\)  are disjoint (indeed, a line \(l^{\prime }\in L(p)\cap L(p^{\prime })\)  must intersect \(N_{\delta }(l)\)  in an interval of length \(\ge \delta ^{1-\varepsilon }\rho ^{-1}\) , but this is impossible since the slopes of \(l\)  and \(l^{\prime }\)  differ by at least \(\delta ^{-T}\rho \ge \delta ^{-\varepsilon }\rho )\) . Thus
\(\# \bigcup _{p\in P^{\prime }}L(p) \gtrapprox (\#P^{\prime })\prod _{i=j}^{k-1}M_i\gtrapprox \rho ^{-\alpha }\prod _{i=j}^{k-1}M_i.\) 
The lines in the above set satisfy \(\operatorname{dist}(\iota (l),\iota (l^{\prime }))\le \rho \) , and thus
\(\prod _{i=j}^{k-1}M_i \lessapprox (\#P^{\prime })^{-1}\#(B(x,\rho )\cap \iota (L_1))\lessapprox (\rho /\delta )^{-\alpha }(\rho /\delta )^{2\alpha }=(\rho /\delta )^\alpha .\) 
Selecting \(C_0\)  sufficiently large based on the implicit constants in the above quasi-inequality (these constants in turn depend only on \(\alpha \) ), we conclude that (REF ) holds for \(j\) . This completes the induction step.
By (REF ), if \(p\in P_2\)  and \(J\subset S^1\)  is an interval whose image \(J^{\prime }\subset [0,1]\)  is a \(2^T\) -adic interval, then
\(\#\lbrace l\in I_2(p)\colon \operatorname{slope}(l)\in J\rbrace \lessapprox (|J|/\delta )^\alpha .\) 
Now, let \(v\in S^1\)  and let \(\delta \le r\le 1\) . Then there is an index \(j\)  so that \(2^{-jT}\le r < 2^{-(j-1)T}\le \delta ^{-\varepsilon }2^{-jT}\) , and thus the set of unit vectors \(\lbrace v^{\prime }\in S^1\colon \angle (v^{\prime },l)\le r\rbrace \)  correspond to a set of slopes \(m\)  that can be covered by a union of \(\approx 1\)  intervals of length \(2^{-jT}\) . We conclude that
\(\#\lbrace l\in I_2(p)\colon \angle (l,v)\le r\rbrace \lessapprox (r/\delta )^\alpha .\) 
Our incidence arrangement \((P_2,L_1,I_2)\)  now satisfies Properties (B), (C), (D), and (F) from Proposition REF  (with \(\lessapprox \)  in place of \(\le \delta ^{-\varepsilon }\)  in some instances), as well as the estimate \(\#I_3\approx \delta ^{-3\alpha }\) .

Robust transversality of lines through a point.
Our goal in this section is to find a set of incidences \(I_3\subset I_2\)  with \(\#I_3\approx \delta ^{-3\alpha }\)  that continues to satisfy Hypotheses (B), (C), (D), and (F), but which also satisfies a re-scaled version of Property (E).
Let \(j_0\ge 0\)  be an index so that \(M_{j_0}>100\) , and \(M_j\le 100\)  for each \(j=0,\ldots ,j_0-1\) ; we can assume that such an index must exist, since otherwise we would have \(\#I_2(p)=\#A_p^{\prime } = \prod _{j=0}^{k-1}M_j \le \prod _{j=0}^{k-1}100 \le 100^{1/\varepsilon }\)  for each \(p\in P_2\) . But then \(\#P_2 \ge \#I_2 / 100^{1/\varepsilon }\gtrapprox \delta ^{-3}\) , which would establish (REF ).
For each \(p\in P_2\) , define \(A_p^{\prime \prime }=A_p^{\prime }\cap J\) , where \(J\)  is a \(2^T\) -adic interval of length \(2^{-j_0T}\)  that intersects \(A_p^{\prime \prime }.\)  By construction we have
\(\#A_p^{\prime \prime }= (N_{0}N_1\cdots N_{j_0-1})^{-1}\#A_p^{\prime } \gtrsim \#A_p^{\prime },\) 
and for any \(2^T\) -adic interval \(J^{\prime }\subset J\)  of length \(2^{-(j_0+1)T}\) , we have
\(\#(A_p^{\prime \prime }\cap J^{\prime })\le \frac{1}{100}\#A_p^{\prime \prime }.\) 
Let \(I_3\subset I_2\)  consist of those pairs \((p,l)\in I_2\)  for which \(\operatorname{slope}(l)\in A_p^{\prime \prime }\) . Define \(s = 2^{-j_0T}\) . We record the following properties of this incidence arrangement.

Properties (B) (with \(V=1\) ) and (C) continue to hold for \((P_2,L_1,I_3)\) , and \(\#I_3\gtrapprox \delta ^{-3}\) .

(REF ) (and thus Property (F)) continues to hold with \(I_3\)  in place of \(I_2\)

For each \(p\in P_2\) , \(\#I_3(p) = \#I_3 (\#P_2)^{-1},\)  and thus Property (D) continues to hold.

For each \(p\in P_2,\)  the set \(\lbrace  \operatorname{slope}(l)\colon l\in I_3(p)\rbrace \subset [0,1]\)  is contained in an interval of length \(s\) . If \(J\subset [0,1]\)  is an interval of length \(\delta ^{\varepsilon }s\) , then
\(\#\lbrace  l\in I_3(p)\colon \operatorname{slope}(l)\in J\rbrace  \le \frac{1}{20}\#I_3(p).\) 
(Indeed, the slopes of the lines in the above set correspond to points in \(A_p^{\prime \prime }\)  that are contained in a union of at most two \(2^T\) -adic intervals of length \(2^{-(j_0+1)T}\) ).

For each \(p\in P_2\) , let \(v_p\in S^1\)  be a vector of the form \((1,m)/\sqrt{1+m^2}\) , where \(m\)  is the midpoint of the interval containing the set of slopes \(\lbrace  \operatorname{slope}(l)\colon l\in I_3(p)\rbrace \) . We can suppose \(v_p=(\cos \theta ,\sin \theta )\) , with \(\theta \in [0,\pi /4]\) .

Partitioning into rectangles.
Let \(\mathcal {Z}_0=([0,1]\times [-1,1])\cap (\frac{s}{2}\mathbb {Z})^2\) . For each \(z=(m,b)\in \mathcal {Z}_0\) , let \(l_z\)  be the line \(y=mx+b\) , and define
\(P_2^z = \lbrace p\in P_2\colon p\in N_s(l_z),\ \angle (v_p, l_z)\le s/2\rbrace ,\quad L_1^z = \bigcup _{p\in P_2^z}I_3(p).\) 
We claim that
\(P_2 = \bigcup _{z\in \mathcal {Z}_0}P_2^z,\) 
and each \(p\in P_2\)  is contained in at most 9 sets in the above union. Indeed, for each \(p\in P_2\) , there exists at least one, and at most three \(m\in [0,1]\cap (\frac{r}{2}\mathbb {Z})\)  with the following property: if \(v_m\)  is the direction of the vector \((1,m)\) , then \(\angle (v_p, v_m)\le s/2\) . Once this choice of \(m\)  has been fixed, there is at least one, and at most three \(b\in [-1, 1]\cap (\frac{r}{2}\mathbb {Z})\)  with the following property: if \(z = (m, b)\) , then \(p\in N_s(l_z)\) .
Observe as well that each \(l\in L_1\)  can be contained in at most 100 sets of the form \(L_1^z\) : if \(l\in L_1^z\)  then \(l\in I_3(p)\)  for some \(p\in P_2^z\) , which implies \(\angle (l,l_z)\le \angle (l, v_p)+\angle (v_p,l_z)\le s\) . Next, since \(p\in N_{\delta }(l)\cap N_s(l_z)\cap [0,1]^2\) , we have that \(\operatorname{dist}(l\cap [0,1]^2,l_z)\le s+\delta \) . But this implies that \(\operatorname{dist}(\iota (l),\iota (l_z))\le 3s\) . The result now follows from the fact that \(\mathcal {Z}_2\)  is \(s/2\) -separated.
Thus if we define \(I_3^z = I \cap (P_2^z\times L_1^z)\) , then \(I_3 = \bigcup _{z\in \mathcal {Z}_0}I_3^z\) , and each \((p,l)\in I_3\)  is contained in at most 9 sets in the above union. By refining \(\mathcal {Z}_0\)  by a factor of at most 9, we can suppose that the sets \(\lbrace P_2^z\rbrace \) , \(\lbrace L_1^z\rbrace \) , and thus \(\lbrace \mathcal {I}_3^z\rbrace \)  are disjoint. Using dyadic pigeonholing, select \(\mathcal {Z}\subset \mathcal {Z}_0\)  so that the sets \(\lbrace P_2^z\rbrace _{z\in \mathcal {Z}}\) , \(\lbrace L_1^z\rbrace _{z\in \mathcal {Z}}\)  (and thus \(\lbrace \mathcal {I}_3^z\rbrace _{z\in \mathcal {Z}}\) ) are disjoint,
\(\sum _{z\in \mathcal {Z}} \#I_3^z\gtrsim \#I_3,\) 
and \(\#P_2^z\)  (and hence \(\#I_3^z\) ) is approximately the same (up to a multiplicative factor of 2) for each \(z\in \mathcal {Z}\) . In particular,
\(\#P_2^z \approx (\#P_2)/(\#\mathcal {Z}),\quad \textrm {and}\quad \# I_3^z\approx (\#I_3)/(\#\mathcal {Z})\quad \textrm {for each}\ z\in \mathcal {Z}.\) 
Geometrically, we have partitioned (a large portion of) the arrangement \((P_2, L_1, I_3)\)  into sub-arrangements of the form \((P_2^z, L_1^z, I_3^z)\) , each of which is contained in the region \([0,1]^2\cap N_{3s}(l_z)\) , and each of which contributes a roughly equal number of incidences. Our next step is to analyze each of these sub-arrangements.

Averaging over segments.
For the remainder of our arguments we will fix a choice of \(z\in \mathcal {Z}\)  with
\(\#L_1^z \gtrapprox \#L_1(\# \mathcal {Z})^{-1}\gtrapprox \delta ^{-2\alpha }(\# \mathcal {Z})^{-1}.\) 
Let \(P=P_2^z,\)  \(L=L_1^z\)  and \(I=I_3^z\) . Let \(\mathcal {S}\)  be the set of \(2^T\) -adic squares in \([0,1)^2\)  of side-length \(2^{ (j_0-2-k)T}\)  (note that \(2^{ (j_0-2-k)T}\approx \delta /s\) ). We claim that the arrangement \((P,L,I)\)  has the following properties: If \(S\in \mathcal {S}\)  and if \(p\in P\cap S\) , then
(i):
There is exactly one line \(l\in I(p)\)  that contributes points to the square \(S\) .
(ii):
Every other line \(l^{\prime }\in I(p)\)  is dominated by \(l\)  on some square \(S^{\prime }\supset S\) .

Item (i) holds in general (see the discussion following (REF )). For Item (ii), since \(l^{\prime }\in I(p)\)  and \(l^{\prime }\ne l\) , we have that \(l\)  contributed points in the square \(S\)  containing \(p\) , and \(l^{\prime }\)  did not contribute points in \(S\) . Let \(S^{\prime }\)  be the largest \(2^T\) -adic square containing \(p\)  so that \(l^{\prime }\)  was dominated by \(l\)  on \(S^{\prime }\)  (the square \(S^{\prime }\)  with this property is necessarily unique). We will show that \(S^{\prime }\supset S\) . If \(S^{\prime }\supsetneq S\)  then we are done. If not, then at the step in the construction of \((P_1,L_1,I_1)\)  when \(l^{\prime }\)  was processed, the square \(S\)  would not be contained in a (larger) dominated square. But we also have that
\(\operatorname{dist}(l \cap S, l^{\prime }\cap S)\le \operatorname{dist}(l \cap S^{\prime }, l^{\prime }\cap S^{\prime })\le (\sqrt{2})2^{-kT},\) 
and
\(\angle (l,l^{\prime })\le \angle (l,l_z)+\angle (l_z,l^{\prime })\le 2s = 2\cdot 2^{-j_0T} \le (\frac{\pi }{2})2^{ ([k -(j_0-2) ] - k) T},\) 
which implies that \(l\)  is dominated by \(l^{\prime }\)  on square \(S\) .
Next, observe that if \(l,l^{\prime }\)  both contribute points to \(S\) , then since \(\angle (l,l^{\prime })\le (\frac{\pi }{2})2^{((j_0+1)-k)T}\) , we must have \(\operatorname{dist}(l \cap S, l^{\prime }\cap S)>(\sqrt{2})2^{-kT}.\)
Let \(L(S)\subset L\)  be the set of lines that contribute points to the square \(S\) . Then the above discussion shows that if \(l,l^{\prime }\in L(s)\)  are distinct, then the sets \(S\cap I(l)\)  and \(S\cap I(l^{\prime })\)  are disjoint. We have
\(P = \bigsqcup _{S\in \mathcal {S}} \bigsqcup _{l\in L(S)} (S\cap I(l)).\) 
Since each set \(P_l\)  is a \((\delta ,\alpha )_1\) -set, each of the sets \((S\cap I(l))\)  have cardinality at most
\(\delta ^{-\varepsilon }\Big (\frac{2^{ (j_0-2-k)T}}{\delta }\Big )^\alpha \lessapprox \Big (\frac{\delta /s}{\delta }\Big )^\alpha = s^{-\alpha }\) 
Thus by dyadic pigeonholing, there is a number \(M\)  with
\(1\le M\lessapprox s^{-\alpha }\) 
so that
\(\#P \approx \sum _{S\in \mathcal {S}} \sum _{\begin{array}{c}l\in L(S)\\ M\le \#(S\cap I(l))<2M\end{array}} \#(S\cap I(l)).\) 
Choose \(Q\subset P\)  by selecting one point from \(S\cap I(l)\)  for each square \(S\in \mathcal {S}\)  and each \(l\in L(S)\)  in the above sum. Define \(I_Q = I\cap (Q\times L)\) . In particular, (REF ) and (REF ) continue to hold with \(Q\)  in place of \(P\) , and \(I_Q\)  in place of \(I_2\)  or \(I_3\) . We have \(\#I_Q\gtrapprox M^{-1}\delta ^{-3\alpha }\) .

Rescaling
Define \(\tilde{\delta }=\delta /s\) . Let \(T\colon \mathbb {R}^2\rightarrow \mathbb {R}^2\)  be a translation in the direction \(\operatorname{dir}(l_z)^\perp \)  that sends \(l_z\)  to a line through the origin, composed with an anisotropic dilation by \(s^{-1}\)  in the direction \(\operatorname{dir}(l_z)^\perp \) ; the image of \(N_{s}(l)\cap [0,1]^2\)  under this linear map is contained in \(B(0, 10)\) , and it has volume \(\sim 1\) . Let \(\tilde{Q}\)  (resp. \(\tilde{L}\) ) be the image of \(Q\)  (resp. \(L\) ) under \(T\) . Let \(\tilde{I}_Q\subset \tilde{Q}\times \tilde{L}\)  be the incidence relation corresponding to \(I_Q\) . If \((\tilde{q},\tilde{l})\in \tilde{I}_Q\)  then \(\tilde{q}\in N_{\tilde{\delta }}(\tilde{l})\) .
If \(S\in \mathcal {S}\)  and \(l,l^{\prime }\in L(S)\) , then \(T(l\cap S)\)  and \(T(l^{\prime }\cap S)\)  are \(\gtrapprox \delta /s=\tilde{\delta }\)  separated. Thus for each such square \(S\) , \(\tilde{Q} \cap T(S) = T(Q \cap S)\)  is \(\gtrapprox \tilde{\delta }\) -separated. But since the squares in \(\mathcal {S}\)  have side-length \(\approx \tilde{\delta }\) , each ball \(B\subset \mathbb {R}^2\)  of radius \(\tilde{\delta }\)  intersect at most \(\approx 1\)  sets of the form \(\lbrace T(S)\colon S\in \mathcal {S}\rbrace \) . We conclude that \(\#(\tilde{Q}\cap B)\lessapprox 1\)  for each ball \(B\subset \mathbb {R}^2\)  of radius \(\tilde{\delta }\) . Refining the set \(\tilde{Q}\)  by a \(\approx 1\)  factor, we can suppose that \(\tilde{Q}\)  is \(\tilde{\delta }\) -separated; this refinement does not affect the validity of any of our previous estimates.
Define \(V=\min (1, (s^\alpha M)^{-1})\) ; by (REF ) we have \(V\approx (s^\alpha M)^{-1}\) , and thus \(M^{-1}\delta ^{-\alpha } \approx V\tilde{\delta }^{-\alpha }\) . We will record the following properties of \(\tilde{Q},\ \tilde{L}\) , and \(\tilde{I}\) .
\(&\#P\gtrsim M\#\tilde{Q},\\&\#\tilde{I}_Q(q) = (\#\tilde{I})(\#\tilde{Q})^{-1}\quad \textrm {for each}\ q\in \tilde{Q},\\&\#\tilde{I}_Q(l)\lessapprox M^{-1}\delta ^{-\alpha }\approx \tilde{\delta }^{-\alpha }V\quad \textrm {for each}\ l\in \tilde{L},\\&\#\tilde{I} \gtrapprox M^{-1}(\delta ^{-\alpha }\#L^{\prime })\gtrapprox \tilde{\delta }^{-\alpha }(\#\tilde{L})V.\) 

() shows that \((\tilde{Q}, \tilde{L}, \tilde{I})\)  satisfies Property (D) from Proposition REF . For each \(l\in \tilde{L}\)  and each ball \(B(r,x)\) , with \(r\ge \tilde{\delta }\) ,
\(\#\lbrace \tilde{q}\in \tilde{Q}\cap B(t,x)\colon (\tilde{q},\tilde{l})\in \tilde{I}\rbrace \le M^{-1}\delta ^{-\varepsilon }(r/\delta )^\alpha \approx (r/\tilde{\delta })^\alpha V,\) 
c.f. Property (B) from Proposition REF . For each ball \(B(t,x)\) , the set of lines
\(\lbrace l\in L\colon \iota (T(l))\in B\rbrace =\lbrace T^{-1}(l)\colon \iota (l)\in B \rbrace \) 
is contained in a ball \(B^{\prime }\)  of radius \(\approx st\) . Thus
\(\# (\iota (\tilde{L}) \cap B)\le \delta ^{-\varepsilon }(st/\delta )^{2\alpha }=\delta ^{-\varepsilon }(t/\tilde{\delta })^{2\alpha }.\) 
c.f. Property (C) from Proposition REF .
Next, for each \(q\in \mathcal {Q}\)  we have the following analogue of (REF ): for each \(v\in S^1\)  and each \(r\ge \tilde{\delta }\) ,
\(\#\lbrace \tilde{l} \in \tilde{I}_Q(\tilde{q})\colon \angle (\tilde{l},v)\le r\rbrace \lessapprox (sr/\delta )^\alpha \le (r/\delta )^\alpha .\) 
Indeed, (REF ) follows from (REF ), since if \(q=T^{-1}(\tilde{q}),\)  then \(\tilde{I}_Q(\tilde{q})=\lbrace T(l)\colon l\in I_Q(q)\rbrace \) . Since the latter set of lines obeys (REF ), the former obeys (REF ). c.f. Property (F) from Proposition  REF
Finally, for each \(\tilde{q}\in \tilde{Q}\)  we have the following analogue of (REF ): If \(v\in S^1\) , then
\(\#\lbrace l\in \tilde{I}_Q(\tilde{q})\colon \angle (l, v)\le \delta ^{\varepsilon }\rbrace  \le \frac{1}{10}\#I_Q(\tilde{q}).\) 
Again, (REF ) follows from (REF ) by the same reasoning as above. Thus \((\tilde{Q}, \tilde{L}, \tilde{I})\)  satisfies Property (E) from Proposition REF .
Remark 5.1 
Observe that arrangement \((\tilde{Q},\tilde{L}, \tilde{I})\)  satisfies Property (E) with room to spare—if we select a smaller set of incidences \(\tilde{I}_Q^{\prime }\subset \tilde{I}_Q\)  so that
\(\#\tilde{I}^{\prime }_Q(\tilde{q}) \ge \frac{1}{4} \#\tilde{I}_Q(\tilde{q}) \quad \textrm {for each}\ q\in \tilde{Q},\) 
then Property (E) will remain true with \(\tilde{I}_Q^{\prime }\)  in place of \(\tilde{I}_Q\) ; we will need this flexibility below.

We have shown that the arrangement \((\tilde{Q}, \tilde{L}, \tilde{I}_Q)\)  satisfies all of the hypotheses of Proposition REF  (or more precisely, an equivalent version of these hypotheses, as discussed in Remark REF ) except Property (A), which is only satisfied on average. To fix this, we shall use the following graph refinement lemma from [6]}.
Lemma 5.2 (Graph refinement) 
Let \(G = (A\sqcup B, E)\)  be a bipartite graph. Then there is a sub-graph \(G^{\prime }=(A^{\prime }\sqcup B^{\prime }, E^{\prime })\)  so that \(\#E^{\prime }\ge \#E/2\) ; each vertex in \(A^{\prime }\)  has degree at least \(\frac{\#E}{4\#A}\) ; and each vertex in \(B^{\prime }\)  has degree at least \(\frac{\#E}{4\#B}\) .

Apply Lemma REF  to \((\tilde{Q}\sqcup \tilde{L},\ \tilde{I}),\)  and let \((\accentset{\approx }{Q}\sqcup \accentset{\approx }{L},\ \accentset{\approx }{I})\)  be the resulting refinement. We can verify that \((\accentset{\approx }{Q}\sqcup \accentset{\approx }{L},\ \accentset{\approx }{I})\)  now satisfies Properties (A) - (F) of Proposition REF . Indeed, by (), each \(\accentset{\approx }{q}\in \accentset{\approx }{Q}\)  satisfies
\(\frac{1}{4}(\#\tilde{I})(\#\tilde{Q})^{-1}\le \# \accentset{\approx }{I}(\accentset{\approx }{q}) \le (\#\tilde{I})(\#\tilde{Q})^{-1},\) 
and hence Property (E) remains true (see the discussion in Remark REF ). (REF ), and thus Property (F) remains true for similar reasons. Similarly for Property (D).
Properties (B) and (C) remain true because \((\accentset{\approx }{Q}\sqcup \accentset{\approx }{L},\ \accentset{\approx }{I})\)  is a refinement of \((\tilde{Q}\sqcup \tilde{L},\ \tilde{I})\) . Finally, Property (A) holds because by () and (), for each \(\accentset{\approx }{l}\in \accentset{\approx }{L}\)  we have
\(\tilde{\delta }^{-\alpha }V \lessapprox \frac{1}{4}(\#\tilde{I})(\#\tilde{L})^{-1}\le \#\accentset{\approx }{I}(\accentset{\approx }{l}) \lessapprox \tilde{\delta }^{-\alpha }V.\) 
Note as well that (REF ), () and () imply that
\(\#\accentset{\approx }{L}\gtrapprox \delta ^{-2\alpha }(\#\mathcal {Z})^{-1}.\) 
Applying Proposition REF , we conclude that
\(\# \tilde{Q} \ge \#\accentset{\approx }{Q} \gtrapprox \tilde{\delta }^{-2\alpha -c(\alpha )}(\tilde{\delta }^{2\alpha }\#\accentset{\approx }{L})V^{c(\alpha )/\alpha }\gtrapprox \tilde{\delta }^{-2\alpha -c(\alpha )}(\tilde{\delta }^{2\alpha }\delta ^{-2\alpha }(\#\mathcal {Z})^{-1})V^{c(\alpha )/\alpha },\) 
where the final inequality used (REF ). And hence by (REF ) and (REF ),
\(\begin{split}\#P_2 &\gtrapprox (\#\mathcal {Z})(\# P)\\& \gtrapprox (\#\mathcal {Z}) M \tilde{\delta }^{-2\alpha -c(\alpha )}(\tilde{\delta }^{2\alpha }\delta ^{-2\alpha }(\#\mathcal {Z})^{-1})V^{c(\alpha )/\alpha }\\& \gtrapprox M (\delta /s)^{-2\alpha -c(\alpha )}(s^{-2\alpha })V^{c(\alpha )/\alpha }\\& \gtrapprox M \delta ^{-2\alpha -c(\alpha )} s^{c(\alpha )}(s^\alpha M)^{-c(\alpha )/\alpha }\\& \gtrapprox \delta ^{-2\alpha -c(\alpha )} M^{1-c(\alpha )/\alpha }\\& \ge \delta ^{-2\alpha -c(\alpha )}.\end{split}\) 
Since \(P_2\subset P_1\) , we obtain (REF ).

Proof of Proposition  REF
In this section, we will sketch the proof of Proposition REF . As discussed in Section , this proof does not contain any new ideas; it is simply the proof of Theorem 1.1 from [2]}, except we keep track of the dependence on \(K_0,\ldots ,K_4\) . We also preserve two distinct estimates that were merged too early in [2]}, which allows for a slightly stronger final bound. The proof sketch presented here is not self contained—the reader should compare with Theorem 1.1 from [2]}.
To prove Proposition REF , it suffices to establish the estimate
\(K_1^6 K_2^{6+\alpha } K_3^{2(9+4\alpha )} K_4^{4(2+\alpha )} \gtrapprox \delta ^{-\alpha (1-\alpha )}.\) 
To begin, we will record some facts about the set \(A\)  that were proved in [2]}. For comparison, the set \(A\)  in the statement of Proposition REF  corresponds to the set \(A^{\prime }\)  from [2]} (In [2]}, \(A^{\prime }\subset A\)  was selected to that \(\mathcal {E}_{\rho }(A^{\prime }+A^{\prime })\le K_3\mathcal {E}_{\rho }(A^{\prime })\)  for all \(\delta \le \rho \le 1\) . In Proposition REF , this property already holds for \(A\) ).
The authors in [2]} use their assumed upper bound on \(\mathcal {E}_{\delta }(A)\)  to obtain the following lower bound on the multiplicative energy of \(A\) :
\(\sum _{a,b\in A}\#\big ((a.A)_{\delta ^+} \cap (b.A)_{\delta ^+}\big )\gtrsim K_4^{-1}(\#A)^3.\) 
We claim that our hypothesis \(\mathcal {E}_{\delta }^\times (A,A)\ge K_4^{-1}(\#A)^3\)  implies the analogous statement, with \(\delta \)  replaced by \(2\delta \) .
Indeed, for each \(a,b\in A\) , select \(a^{\prime },b^{\prime }\in A_{\delta ^\times }\)  with \(|a-a^{\prime }|< 2\delta ,\ |b-b^{\prime }|<2\delta \) . Then
\(\#\big ((a.A)_{2\delta ^+} \cap (b.A)_{2\delta ^+}\big )\gtrsim \#\big ( (a^{\prime }. A_{\delta ^\times })\cap (b^{\prime }. A_{\delta ^\times }).\) 
By (REF ) (with \(2\delta \)  in place of \(\delta \) ) and pigeonholing, we conclude that there exists \(b\in A\)  so that
\(\sum _{a\in A}\#\big ((a.A)_{2\delta ^+} \cap (b.A)_{2\delta ^+}\big )\gtrsim K_4^{-1}(\#A)^2.\) 
(REF ) is the analogue of (8) from [2]}. From here, the argument proceeds in a almost identical fashion, except we track the dependence on \(K_1,K_2,K_3,K_4\) . We will briefly highlight a few of the key estimates from [2]}, and give the analogous statements.
The authors in [2]} select a number \(K_4^{-1}\le \rho \le 1\)  and a set \(\bar{A}\subset A\) , with \(\#\bar{A}\gtrsim |\log \delta |^{-1}(\#A)\) , so that
\(\#\big ( (aA)_{2\delta ^+}\cap (bA)_{2\delta ^+}\big )\sim (\# A)K_4^{-1}\rho ^{-1}\quad \textrm {for each}\ a\in \bar{A}.\) 
They then select a set \(A_1\subset \bar{A}\)  with \(\#A_1\gtrapprox \#\bar{A}\) , so that \(A_1+A_1\)  has small \(\rho \) -covering number for all \(\delta \le \rho \le 1\) . The key estimates we need are the following analogues of Lemmas 3.6 and 3.7 from [2]}. Let \(a_1,a_2,b_2,b_2\in A_1\)  and let \(d_1=a_1-b_1,\)  \(d_2 = a_2-b_2\) . Then
\(\mathcal {E}_{\delta }(d_1A+ d_2A)\lessapprox K_1K_2K_3^{8}K_4^4\rho ^{4} \max (|d_1|,|d_2|)^{\alpha }\#(A),\) 
and for each \(k\ge 2\)  there is a set \(A_2\subset A_1\) , with \(\#A_2\ge \frac{1}{4}\#A_1\) , so that
\(|d_1A_2+ \underbrace{d_2A_2 +\ldots + d_2A_2}_{\text{$k$ times}}|\lessapprox K_1K_2K_3^{7+k}K_4^4\rho ^{5-k} \max (|d_1|,|d_2|)^{\alpha }\#(A).\) 
Next, let \(s\)  and \(\gamma \)  be parameters that we will specify below. Proceeding as in [2]}, we define
\(B = \Big \lbrace \frac{a_1-a_2}{a_3-a_4}\colon a_i\in A_1,\ |a_3-a_4|>\delta ^\gamma \Big \rbrace .\) 
Lemma 4.1 from [2]} says that either there is a point \(b\in B\cap [0,1]\)  so that both \(b/2\)  and \((b+1)/2\)  have distance at least \(s\)  from \(B\) , or the \(s\) -covering number of \(B\cap [0,1]\)  is comparable to \(s^{-1}\) . The former situation is called the “gap case,” while the latter is called the “dense case.”
Our arguments in both of these cases mirror those of [2]}, with (REF ) and (REF ) in place of Lemmas 3.6 and 3.7. The arguments from [2]} show that in the dense case, we must have either
\(K_2K_3^8K_4^4 \gtrapprox \delta ^{\alpha }s^{-1}\quad \textrm {and/or}\quad K_1^3K_2K_3^8K_4^4 \gtrapprox \delta ^{-\gamma \alpha },\) 
while in the gap case we must have
\(K_1^3K_2K_3^{10}K_4^4 \gtrapprox \left(\delta ^{\gamma -1}s\right)^{\alpha }.\) 
Until this point, the arguments above have proceeded in parallel to those in [2]}. The improvement over [2]} comes from properly choosing \(s\)  and \(\gamma \) ; we choose \(s\)  and \(\gamma \)  so that
\(s = \left(K_1^3K_2^2K_3 \delta ^{3\alpha /2}\right)^{2/(\alpha +2)},\quad \delta ^{\gamma } = K_3^{1/\alpha }\delta ^{1/2}s^{-1/2}.\) 
We can suppose that \(0<s<1\) , since otherwise \(K_1^3K_2^2K_3>\delta ^{-3\alpha /2}\) , which is a stronger estimate than (REF ) (recall that by assumption, \(K_4\ge 1\) ). Similarly, we can suppose that \(0<\delta ^\gamma <1\)  (and hence \(\gamma >0\) ), since otherwise \(K_3^{4/\alpha }\delta ^{2-2\alpha } \ge \left(K_1^6K_2^4\right)\ge 1,\)  and hence \(K_3\ge \delta ^{-\frac{\alpha (2-\alpha )}{4}},\)  which again is a stronger estimate than (REF ) (recall that by assumption, \(K_1,K_2K_4\ge 1\)  and \(\alpha \in (0,1)\) ).
Since at least one of the three inequalities from (REF ) and (REF ) must hold, we can verify that with this choice of \(s\)  and \(\gamma \) , (REF ) and (REF ) imply (REF ).
